Download Modeling Root Zone Effects on Preferred Pathways for the Passive

ORIGINAL RESEARCH
published: 23 June 2016
doi: 10.3389/fpls.2016.00914
Modeling Root Zone Effects on
Preferred Pathways for the Passive
Transport of Ions and Water in Plant
Roots
Kylie J. Foster and Stanley J. Miklavcic *
Phenomics and Bioinformatics Research Centre, School of Information Technology and Mathematical Sciences, University of
South Australia, Mawson Lakes, SA, Australia
Edited by:
Janin Riedelsberger,
University of Talca, Chile
Reviewed by:
Ingo Dreyer,
University of Talca, Chile
Xiaoxian Zhang,
Rothamsted Research, UK
*Correspondence:
Stanley J. Miklavcic
[email protected]
Specialty section:
This article was submitted to
Plant Physiology,
a section of the journal
Frontiers in Plant Science
Received: 03 March 2016
Accepted: 09 June 2016
Published: 23 June 2016
Citation:
Foster KJ and Miklavcic SJ (2016)
Modeling Root Zone Effects on
Preferred Pathways for the Passive
Transport of Ions and Water in Plant
Roots. Front. Plant Sci. 7:914.
doi: 10.3389/fpls.2016.00914
We extend a model of ion and water transport through a root to describe transport along
and through a root exhibiting a complexity of differentiation zones. Attention is focused
on convective and diffusive transport, both radially and longitudinally, through different
root tissue types (radial differentiation) and root developmental zones (longitudinal
differentiation). Model transport parameters are selected to mimic the relative abilities
of the different tissues and developmental zones to transport water and ions. For each
transport scenario in this extensive simulations study, we quantify the optimal 3D flow
path taken by water and ions, in response to internal barriers such as the Casparian strip
and suberin lamellae. We present and discuss both transient and steady state results
of ion concentrations as well as ion and water fluxes. We find that the peak in passive
uptake of ions and water occurs at the start of the differentiation zone. In addition, our
results show that the level of transpiration has a significant impact on the distribution of
ions within the root as well as the rate of ion and water uptake in the differentiation zone,
while not impacting on transport in the elongation zone. From our model results we infer
information about the active transport of ions in the different developmental zones. In
particular, our results suggest that any uptake measured in the elongation zone under
steady state conditions is likely to be due to active transport.
Keywords: Casparian strip, suberin lamellae, elongation zone, differentiation zone, transpiration, salt uptake
1. INTRODUCTION
Plant roots are responsible for the uptake of water and nutrients, as well as any potentially toxic
ions, from the soil. This uptake varies along the longitudinal axis of roots, particularly across the
different developmental zones, as a result of both changes in internal root anatomy (e.g., Zhou
et al., 2011) and changes in the functional expression of transporters. There has been extensive
investigation into the identification of ion transporters and their role in the uptake and transport
of ions in plant roots (Barberon and Geldner, 2014). However, understanding how changes in
root anatomy influence the transport of ions and water is also important for improving our
understanding of root function. In this paper we utilize mathematical modeling to explore the
effects of cell differentiation on the preferred pathways for the passive uptake of ions and water
by plant roots. This model is an extension of previous efforts (Foster and Miklavcic, 2013, 2014),
which have already highlighted the intrinsic coupling of ions in their transport through the root.
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are present to those where the barriers are absent (e.g., Foster and
Miklavcic, 2014; Sakurai et al., 2015). Modeling also allows us to
determine the relative contributions of different driving forces,
such as diffusion and convection, to the overall transport process.
Finally, it is relevant to consider how much of the observed
ion uptake and concentration distributions can be explained
by passive processes. This would be difficult or impossible to
explore experimentally, whereas modeling can simulate scenarios
in which uptake in a root occurs by passive processes without the
interference of active transporters.
Previous simulations of solute uptake by plant roots have
typically investigated radial, rather than axial, variations in root
transport properties. For example, Claus et al. (2013) and Sakurai
et al. (2015) developed models of the uptake of zinc and silicon,
respectively, in the DZ. Their efforts focused on the radial
distribution of these solutes (including the effects of apoplastic
transport barriers) and as a result they did not consider any
variations in transport properties along the longitudinal axis
of the root. In contrast, Shimotohno et al. (2015) modeled
the uptake and spatial distribution of boron in a plant root,
including axial variation in the model root transport properties to
simulate different root developmental zones. The emphasis of the
simulations conducted by Shimotohno et al. (2015), however, was
on the root tip and hence the effects of endodermal barriers and
functional xylem were not simulated. All of the transport models
described above included transport via membrane transporters.
However, none of these models simulated transport in a plant
root which had a zone of undifferentiated cells as well as a zone in
which functional xylem and endodermal barriers were present.
Convection is likely to be an important driving force for
ion transport, particularly in the DZ where functional xylem is
present. However, convection is often not modeled in root ion
transport simulations. An exception is the zinc model developed
by Claus et al. (2013) which included convection in the symplast
via a constant water flow rate. Claus et al. (2013) identified that
this convection was an important factor affecting the radial zinc
concentration pattern. However, this model did not incorporate
all of the interactions between solutes and water. In particular, the
osmotic effects of ion concentrations on water transport were not
modeled.
In this study we extend our compartmental model of ion
and water transport in a plant root (Foster and Miklavcic,
2013, 2014) to simulate different developmental zones along the
longitudinal axis of the root. We include both the EZ and the DZ,
incorporating the transition from non-functional to functional
xylem and the development of the endodermal barriers. In
contrast to the ion transport models discussed above, we focus on
passive transport and include interactions between ion and water
transport. As assumed previously (Foster and Miklavcic, 2013,
2014), we do not explicitly consider the apoplastic and symplastic
pathways separately. Instead, the transport parameters, fluxes
and concentrations are a composite representation of both
pathways.
Using our model, we identify that the main steady state,
passive, uptake of ions and water occurs at the start of the DZ. In
contrast, the passive uptake in the EZ at steady state is negligible
due to the insignificant level of convection in this region. From
Plant roots can be divided longitudinally into anatomically
distinct developmental zones, each of which have different
abilities to take up and transport ions and water. For example,
Arabidopsis thaliana primary roots consist of three distinct zones
(Dolan et al., 1993): the meristematic zone, followed by the
elongation zone (EZ) and then the differentiation zone (DZ).
In the EZ the cells lengthen in preparation for differentiation,
while in the DZ the cells become fully differentiated (Dolan et al.,
1993). This process of cell differentiation can significantly affect
the transport of ions and water. For example, in the primary
development stage of the endodermis the Casparian strip (CS)
develops in the cell wall of adjacent endodermal cells. The CS is a
barrier that consists predominantly of lignin (Naseer et al., 2012)
which blocks the passive uptake of ions (and possibly water) via
the non-selective, apoplastic pathway (Geldner, 2013; Barberon
and Geldner, 2014). In addition, the suberin lamellae (SL) forms
in the secondary development stage of the endodermis. The SL
forms between the plasma membrane and the primary cell wall
of the endodermal cells. This barrier eventually covers the entire
cell and inhibits the uptake of water and solutes via transporters
across the plasma membrane of the affected endodermal cells
(Geldner, 2013; Barberon and Geldner, 2014). On the other hand,
the SL does not sever the plasmodesmatal connections between
the cells. Hence, this disruption of the selective pathway forces
solutes and water to enter the symplastic pathway in the outer
regions of the root in order to enter the stele (Geldner, 2013).
Numerous experiments, conducted over many decades, have
investigated the uptake of ions and water along the longitudinal
axis of roots and how this uptake relates to root anatomy (e.g.,
Graham et al., 1974; Lüttge, 1983; Zhou et al., 2011). Similarly,
experimental measurements of how transport properties (such
as hydraulic conductivity) vary across different developmental
zones have been carried out (e.g., Melchior and Steudle, 1993;
Frensch et al., 1996). In addition, experiments have been
conducted to identify the cell-specific distribution of ions
across roots in order to gain information about transport
mechanisms, including the importance of root structures such as
the endodermal barriers as control points in the uptake process
(e.g., Lauchli et al., 2008). An improved understanding of this
uptake has important implications for better understanding root
function, including the transport of nutrients and the response
of plants to various stresses. Clearly, the spatial distribution of
ion transporters has a significant influence on the uptake of
ions along the length of the root as well as on the distribution
of ions within the root. However, the focus of this work is on
the effects of cell differentiation on uptake. Direct experimental
testing of the roles of anatomically distinct root zones has proved
difficult (Barberon and Geldner, 2014). Genetic advances, such
as the recent identification of sgn3, an Arabidopsis mutant with
a nonfunctional CS (Pfister et al., 2014), are important. To
complement these experimental efforts mathematical modeling
can provide an opportunity to examine in detail the spatial
distribution of ion and water fluxes, as well as ion concentrations,
and how these vary over time. Modeling also provides the ability
to explore directly the effects of cell differentiation on ion and
water transport. For example, the role of endodermal barriers can
be examined by comparing model scenarios where the barriers
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be used to examine how much of the observed behavior of ions
in plant roots can be explained by passive processes.
The focus of this report is on the transport of solutes up into
the plant rather than transport of any solutes (such as sucrose)
down into the root. Hence, we assume that the transpiration
stream occurring in the xylem is the dominant function of the
vascular bundle.
this we conclude that any significant uptake in the EZ under
steady state conditions is likely to be driven by active transport. In
addition, we show that the level of transpiration has a significant
effect on the pattern of ion distribution (both radially and axially)
in the DZ and on the rate of ion and water uptake into the
aerial parts of the plant. This has important implications for both
modeling and experimental efforts.
2.2. Forces Driving Ion and Water Transport
2. MODEL OF WATER AND ION
TRANSPORT
In this section we outline the methods used to model ion and
water transport. The detailed model equations are provided in
the Supplementary Material.
We assume that the root consists of rigid and completely
water filled compartments. The radial and axial flow of water
in tissue regions where cell membranes are present is driven
by both hydraulic and osmotic pressure gradients. Hence, the
flow rate of water in these regions is modeled using nonequilibrium thermodynamics (Katchalsky and Curran, 1965). We
use the van’t Hoff relation to calculate the osmotic pressure
in these regions (Katchalsky and Curran, 1965). In contrast,
axial transport in the mature xylem is not interrupted by cell
membranes and hence, the axial flow of water in the xylem is
driven by hydraulic pressure gradients only. This water flow
is modeled as linearly proportional to the hydraulic pressure
gradient, using Darcy’s Law.
The radial and axial transport of ions in the model plant root
is governed by a chemical potential contribution (arising from
concentration differences), an electric field contribution (due to
electric potential differences) and by convection. This is modeled
using the extended Nernst-Planck equation (van der Horst et al.,
1995). The electric potential is determined by solving Poisson’s
equation as described previously (Foster and Miklavcic, 2013,
2014). The transport of ions and water are interdependent due to
both the contribution of water transport to the convection of the
ions as well as the contribution of the ions to the osmotic pressure
driving water transport.
2.1. Model Root Structure
In this compartmental model we simulate the passive uptake
and transport of multiple ions and water through a plant root.
The model root consists of a single, flat ended, right cylinder
(Figure 1A). The anatomy of the root is modeled on the structure
of Arabidopsis primary roots. The model includes radial variation
in transport properties to represent the different tissue types,
as well as longitudinal variation in transport parameters to
represent the different developmental zones (the EZ and DZ).
Due to the compartmental nature of the model, the root is
discretized radially into consecutive annular cylinders which
represent the different tissue types present in Arabidopsis roots:
the epidermis, cortex, endodermis, pericycle and xylem (see
Figure 1). In addition, the model root is further discretized
along the longitudinal axis with the height of each compartment
reflecting the height of an individual cell. Hence, the height of
each compartment varies across the EZ (see Figure 1B).
For simplicity we assume axial symmetry. Therefore, the
features of the model root are independent of the angle around
the central axis, instead they depend only on the radial and
axial coordinates. As a result, any modeled disruption of the
endodermal barriers (e.g., due to the presence of passage
cells) is represented as a disruption of the barrier for the full
circumference of the endodermis at the relevant location along
the length of the root (see Figure 1A).
As an extension of our previous work (Foster and Miklavcic,
2013, 2014) both axial and radial flow of water and solutes is
modeled in all of the root tissues. This allows us to investigate the
effects of changes in root anatomical structure that occur along
the length of the root. To capture the interactions between water
and ion transport, we model their uptake as a coupled system.
Radial and axial flows of water occur due to both osmotic and
hydraulic pressure gradients, while both radial and axial flows
of ions are driven by local concentration differences, electric
field effects and convection. This is a compartmental model
in which the root is divided into model compartments (see
Figures 1B,C), with the hydrostatic pressure and concentration
of each ion assumed to be uniform throughout each model
compartment. The discretization of the root into compartments
is described in further detail in the Supplementary Material.
As this is a composite model water and ion fluxes are not
explicitly separated into apoplastic and symplastic contributions.
Instead, the transport parameters, fluxes and concentrations are
composites of both pathways. This model does not explicitly
include transport via active transporters. To our advantage, it can
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2.3. Model Parameters
Table 1 summarizes the parameter values used in the model
simulations. The boundary conditions are described in Section
2.4.
The values of several model transport parameters were used
to model the anatomical changes that occur across the different
root tissues and root developmental zones. For example, the
increased resistance to transport due to the CS and subsequently
the SL were represented by changing the value of transport
parameters at the endodermis-pericycle interface in the DZ,
without changing the values adopted for the other tissue layers.
In particular, the reduced diffusion due to these barriers was
modeled by decreasing the radial diffusive permeability of
each ion (krad ) by an order of magnitude for each barrier.
Simultaneously, the reduced water transport and convection due
to these barriers was modeled by decreasing the radial water
permeability (Lrad
p ) by an order of magnitude and increasing the
radial reflection coefficient of each ion (σ rad ) by 0.2 units for each
barrier.
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FIGURE 1 | (A) Schematic of the model root showing the radial discretization used to represent the different root tissue types (from outermost region to innermost
region: the epidermis, cortex, endodermis, pericycle, and xylem). The purple and red cylinders show the location of the CS and SL, respectively. (B) Schematic
longitudinal cross-section of the model root. The light gray lines show the radial and axial discretization. The different tissue types are highlighted by different colors,
while the extent of different developmental zones (EZ and DZ) are indicated by arrows. Some of the model boundary conditions are also indicated. For visual clarity,
the z and r axes are scaled differently. (C) Expanded view of a small section of the root cross-section in (B) showing the location of a passage cell. The z and r axes
are here drawn to the same scale (refer to the scale bar) highlighting the actual relative axial to radial proportions. The blue arrows indicate ion fluxes. In (B,C), vertical
purple lines indicate the location of the CS, vertical red lines represent SL and yellow lines represent functional xylem. Purple dashed lines indicate incomplete CS
while yellow dashed lines represent the transition from non-functional to functional xylem. The root dimensions are given in Table 1. The coordinate system (r, z) is as
shown in (A). Gravity force is in the direction of decreasing z.
occasional passage cell (Geldner, 2013). To model this observed
behavior we assumed that the probability of the existence of
a passage cell (i.e., a cell unaffected by SL) at a given location
decreased along the length of the root. To achieve this, we used a
scalar random generator to assign each endodermal
cell as either
The functional CS and xylem were assumed to begin at the
12th cell from the start of the EZ (Naseer et al., 2012). To
represent the observed gradual, patchy appearance of the CS in
the early stages of endodermal cell differentiation (Roppolo et al.,
rad were changed gradually across several
2011) krad , Lrad
p and σ
axial model compartments from values equal to the surrounding
tissues to values representing the fully formed, continuous CS.
This region of developing CS is represented by dashed purple
lines in Figure 1B. Similarly, the development of the xylem
from non-functional to functional xylem in the transition from
the EZ to the DZ was modeled by changing the axial water
permeability (Lax
p ) and the axial reflection coefficient of each ion
(σ ax ) across several axial model compartments (dashed yellow
lines in Figure 1B).
In the early stages of secondary endodermal cell
differentiation, the SL is patchy. As the root matures the SL
becomes a more continuous barrier, with the presence of only an
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j
a passage cell (with a probability of exp − 30 ) or a cell affected
j
by SL (with a probability of 1 − exp − 30 ). Here j refers to the
number of cells from the start of the EZ. The possibility of the
SL first appearing was assumed to occur starting from the 38th
cell from the start of the EZ (Naseer et al., 2012). This process
provided a realistic distribution of passage cells (see Figure 1B).
2.4. Computational Details
The discretization of our model root is discussed in detail in the
Supplementary Material. However, one major difference between
our current model and our previous root model (Foster and
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root (local effects). These non-uniform boundary effects could
be included in future simulations by including the simultaneous
modeling of water and ion transport in the soil. However, the
focus of our work currently is on the response within the root
and hence we have chosen to adopt the simplest of external
conditions. Our model establishes a basic understanding of the
root transport response under optimal external conditions. There
is zero flux of water and ions across the bottom boundary of the
root (as shown in Figure 1B).
The boundary conditions across the top of the root represent
the connectivity of the root to the aerial sections of the plant.
It is clear that the axial flow of water and ions up through
the xylem into the stem should be significant and hence a zero
flux boundary condition across the top of the root would not
be appropriate. However, it is less clear how much flow occurs
axially out of the other tissue types. Hence, we carried out
model simulations for a range of boundary conditions at the top
boundary of the model root. These included: axial flux of water
and ions out of only the xylem with constant concentrations
and hydrostatic pressure at the xylem boundary; axial flux out
of only the xylem with a constant hydrostatic pressure and no
concentration gradients across the xylem boundary; identical
boundary conditions across the top of the root for all tissue
types with constant concentrations and pressure at the boundary;
and identical boundary conditions across the top of the root for
all tissue types with a constant pressure and no concentration
gradients across the boundary. We explored a wide range of
values for the fixed hydrostatic pressure as well as the ion
concentrations at the boundary. The pressure at the top boundary
of the root (Pb ) significantly affected the simulation results. This
is unsurprising since Pb simulates the level of transpiration,
which is one of the key driving forces in our model. However,
the remaining changes to the boundary conditions that were
investigated led to only negligible changes in the simulation
results, with only the fluxes and ion concentrations in the top
three cell layers of the root affected. For the simulations shown
in Section 3 we assume there is zero flux axially out of the outer
four tissue regions at the top boundary of the root and there is a
constant pressure and no concentration gradients across the top
of the xylem (see Figure 1B). In Section 3 we explore the effects
of different Pb values, which represent the effects of different
transpiration conditions.
For the majority of the simulations the root is initially empty
of ions. However, as found in our previous work (Foster and
Miklavcic, 2013) the steady state model results are independent
of the initial conditions. In all instances the root is completely
filled with water.
TABLE 1 | Summary of parameters used and their source.
Parameter
Root hydraulic conductivity, Lp
Value
5.5 × 10−8 ms−1 MPa−1 (Unless otherwise
stated)
Solute permeability, km
3 × 10−9 ms−1 (Unless otherwise stated)
Reflection coefficient, σm
0.5 (Unless otherwise stated)
Axial xylem water permeability, k ax
5 × 10−13 m2
Dynamic viscosity, µ
1.002 × 10−3 Pa s
Water density, ρ
1.0 × 103 kg m−3
Relative permittivity, ǫr
80
Thickness of epidermis
15 µm
Thickness of cortex
20 µm
Thickness of endodermis
10 µm
Thickness of pericycle
5 µm
Radius of xylem
3 µm
Concentration of ions in soil
100 mM
Unless otherwise stated the transport parameters in the radial and axial directions are
assumed to be equal. Root hydraulic conductivity was obtained from Javot et al. (2003),
Ranathunge and Schreiber (2011), and Sutka et al. (2011). Solute permeabilities were
obtained from Ranathunge and Schreiber (2011). Reflection coefficients were obtained
from Boursiac et al. (2005) and Ranathunge and Schreiber (2011). Axial water permeability
in the xylem was calculated using Poiseuille’s Equation. The thicknesses of the root
sections were obtained from Dolan et al. (1993), Scheres et al. (1995), Mattsson et al.
(1999), Casimiro et al. (2003), and Javot et al. (2003).
Miklavcic, 2013, 2014) is that the axial discretization of the root
includes a region of varying compartment height to model the EZ
(see Figure 1B).
The nonlinear, coupled system of ordinary differential
equations, resulting from the model setup described in Section
2.2 and the Supplementary Material, was solved numerically
in MATLAB using the ode15s package. Poisson’s equation was
solved numerically using finite difference methods to determine
ψ as described in Foster and Miklavcic (2014). The hydraulic
pressure in each compartment was determined at each time step
by solving a linear system of equations as described in Foster and
Miklavcic (2014).
The boundary conditions in the soil consist of a linear
hydrostatic hydraulic pressure gradient and a fixed bulk
concentration of four monovalent ions (two monovalent cationanion pairs). The root is assumed to be vertical (see Figure 1),
with the top of the root aligned with the surface of the external
medium. Hence, as a result of the assumption of a hydrostatic
pressure gradient outside the root, the pressure in the external
medium at the top of the root is atmospheric, with this external
pressure increased by ρ × g × L at the bottom of the root (here, ρ
is the density of water, g is the acceleration due to gravity and L is
the length of the root). These ideal boundary conditions assume
that the root is surrounded by an infinite reservoir of both mobile
ions and water and hence can be thought to be representative
of the external conditions in, say, hydroponics experiments. In
contrast, under actual field conditions there is usually a limited
supply of water and solutes in the soil. Deviation from these
ideal conditions may occur in practice for various reasons, such
as non-uniform soil conditions, water depletion in the soil due
to environmental factors (global effects) and uptake into the
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3. RESULTS
The Endodermal Barriers Significantly
Inhibit the Passive Influx of Ions and Water
into the DZ
Figure 2 shows the steady state ion concentration (colormap)
and ion flux (arrows) results for different endodermal barrier
scenarios. In Figure 2A the root is unprotected by endodermal
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state the concentration of ions in the EZ for all four scenarios is
equal to the bulk concentration in the soil.
In contrast, the introduction of the endodermal barriers
has a strong effect on the amount and location of ion and
water uptake in the DZ (see arrows in Figure 2). When there
are no endodermal barriers present, the main uptake of water
and ions occurs throughout the region where the xylem is,
or is close to, fully functional (Figure 2A). However, when
the endodermal barriers are present, the main uptake occurs
closer to the root tip where the CS is not yet fully developed
(giving an average net radial ion flux of 15.19 nmol s−1 m−2
across the root surface in this region at steady state for the
scenario shown in Figure 2D), despite the fact that in this
region the xylem is not fully functional (Figures 2B–D). The
SL substantially blocks water and solute uptake with an average
ion flux of 0.58 nmol s−1 m−2 across the root surface in the
region where the SL is fully developed (excluding passage
barriers. In Figure 2B, the CS is present which blocks the
apoplastic pathway at the endodermis. In Figure 2C, both the CS
and a solid, uninterrupted SL barrier are present. Hence, both
the apoplastic and selective transport pathways are affected at
the endodermis. Finally, Figure 2D represents the most realistic
scenario in which both the CS and the SL are present with a
gradual introduction of the SL and passage cells occurring along
the length of the root. At the passage cell locations the SL is absent
but the CS is still present (Peterson and Enstone, 1996).
In all of the scenarios, there is no significant uptake (i.e., net
radial flux across the root surface) of solutes or water at steady
state in the EZ (see arrows in Figure 2). As there is no functional
xylem in the EZ, there is no convection of ions further up into
the root. The influence of the conductive xylem extends only
a few cells into the EZ, and hence this region is hydraulically
isolated from the rest of the root. Therefore, diffusion is the main
mechanism of ion transport in this region. As a result, at steady
FIGURE 2 | Plots of steady state ion concentrations (colormaps) and ion fluxes (arrows) for four different root structures: (A) no endodermal barriers
present, (B) CS present, (C) CS and SL present, (D) CS and SL with passage cells present. Yellow vertical lines show the presence of functional xylem with
dashed lines near the root tip indicating the gradual increase in conductivity as the xylem develops. Purple vertical lines represent the presence of the CS; dashed lines
near the root tip indicate the gradual appearance of the barrier. Red vertical lines show the location of the SL, with gaps identifying the location of passage cells. The
arrows show the magnitude of net radial solute flux across the root surface (in mol s−1 ), with the same scale used across all four subfigures. These arrows are also
indicative of the corresponding water flow rates. All simulations assume 100 mM of 2 monovalent cations and 2 monovalent anions in the soil (e.g., 100 mM NaCl and
100 mM KNO3 ). All ions are assumed to have identical transport properties and the transport parameters used are provided in Table 1. Refer to Section 2.3 for an
explanation of the simulation of the endodermal barriers and xylem development. These simulations were carried out using Pb = −0.5 MPa.
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cell locations) compared to 6.73 nmol s−1 m−2 in the region
where the CS is fully developed (Figures 2C,D). However, at
the location of passage cells there is increased uptake, with an
average ion flux across the root surface of 5.71 nmol s−1 m−2
at their location (Figure 2D). This enhanced uptake supports
the idea that passage cells function as key entry points into
the stele (Peterson and Enstone, 1996; Andersen et al., 2015).
The role of passage cells is explored further in the following
sections.
In terms of ion concentrations, the main effects of the
introduction of each barrier include an increase in the
concentration gradient across the barrier, as well as a decrease
in the concentration gradient across the tissues inside the barrier
(i.e., the pericycle-xylem interface). Both of these changes result
from relatively large reductions in the concentrations of ions in
the pericycle, with smaller changes in concentrations in the xylem
and endodermis also occurring. The introduction of the first
barrier (the CS) causes the greatest change, with the introduction
of a further barrier (the SL) having a less significant impact.
The results shown in Figure 2 were simulated using a pressure
boundary of −0.5 MPa at the top of the root (see Section 2.4
for further details about the boundary conditions). However,
the discussion in this section is valid at lower transpiration
conditions. The effects of different transpiration conditions are
discussed in the following sections.
FIGURE 3 | Line plot showing the steady state axial flux of ions out of
the top of the root xylem as a function of pressure at the top of the
root, Pb (representing different rates of transpiration), and of the
barriers present in the endodermis (different line types). The scenarios
considered are: no endodermal barriers (solid red line); CS only (dashed blue
line); both CS and SL present with no passage cells (green dotted line); and
both CS and SL with passage cells present (purple dot-dashed line). The
labels indicate the corresponding scenarios in Figure 2. The axial fluxes into
the xylem compartment three cells down from the top of the root were used to
minimize the influence of the flux boundary conditions chosen at the top of the
root. The solute fluxes shown are also indicative of the corresponding water
flow rates. All simulations assume 100 mM of 2 monovalent cations and 2
monovalent anions in the soil. All ions are assumed to have identical transport
properties and the transport parameters used in the simulations are provided
in Table 1.
The Endodermal Barriers Reduce the Rate
of Ion and Water Uptake into the Shoot,
with Greater Reductions at Higher
Transpiration Rates
Considering the relatively small changes in ion concentrations
in the xylem, with the introduction of the endodermal barriers
(Figure 2), it may at first glance appear that these barriers have
minimal impact on the transport occurring within the xylem.
However, other changes in root function occur in response to the
introduction of each barrier. In particular, there are substantial
decreases in both axial ion and water flow rates in the xylem
(Figure 3). Importantly, it is this rate of axial transport out of
the top of the root that determines the amount of ions and water
entering the stem from the root.
The introduction of each endodermal barrier leads to reduced
transport of both ions and water from the root into the aerial
parts of the plant (Figure 3). The effect of each barrier becomes
more significant as the transpiration rate increases (represented
by a more negative Pb ). These results show the compromise
inherent in the function of the endodermal barriers: the entry
of harmful ions into the xylem must be minimized while still
maintaining the uptake of beneficial ions and water. The model
simulations show a small increase in the axial flow rates of
ions and water at the top of the root due to the presence of
passage cells (see Figure 3, purple dot-dashed line vs. green
dotted line), with a higher number of passage cells resulting
in a larger increase (results not shown). These results support
the idea that one function of passage cells is to enhance the
uptake of water and at least some ions (Peterson and Enstone,
1996).
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Delayed Development of CS Relative to
Functional Xylem Significantly Enhances
Passive Ion and Water Uptake
In Arabidopsis primary roots, the xylem has been observed to
appear in conjunction with the CS (Alassimone et al., 2010).
We have thus far assumed that the development of the CS
and functional xylem is coincident along the length of the root
(Figure 2). However, exposure to various stresses (e.g., salinity)
can influence the development of the endodermal barriers
(Enstone et al., 2002) as well as the xylem (Cruz et al., 1992;
Reinhardt and Rost, 1995), altering their position relative to
the root tip and to each other. In Figure 4 we explore the
physiological effects of shifting the first points of development
of the xylem and CS, both together and relative to each other.
Figures 4A–F show the different locations considered, with
vertical lines showing the positions of the CS (purple lines) and
functional xylem (yellow lines). The arrows indicate the ion
fluxes across the root surface, but are also representative of the
water flow rates. Although the results in Figures 4A–F assume
Pb = −0.3 MPa, the qualitative behavior is very similar for other
transpiration conditions; the peak in uptake, however, becomes
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less pronounced as the transpiration rate decreases. Figure 4G
shows the rate of ion transport out of the top of the root xylem for
the scenarios shown in Figures 4A–F for a range of transpiration
conditions. The different positions shown in Figures 4A–F have
only a minimal effect on the concentration of the ions in the root
xylem at the top of the model root (results not shown).
A key result is that it is the relative location of the first points
of development of the xylem and the CS that has the largest
effect on the axial transport of water and ions up out of the
root (Figure 4G). If any length of xylem is unprotected by the
endodermal barrier, due to either delayed development of the CS
(Figure 4A) or early maturation of the xylem (Figure 4B), there
is increased transport of water and ions radially into the root
and hence out of the xylem at the top of the root (green lines
in Figure 4G). In contrast, if the CS develops before the xylem
(Figures 4E,F) this transport is reduced (red lines in Figure 4G).
These results suggest that experiments examining the
development of endodermal barriers could also quantify the
development of the xylem and the relative positions of the two.
In particular, examining the relative development of the CS and
xylem and how this correlates with applied stresses could provide
useful insights into the stress response of plants. For example,
salt stress has been observed to result in the endodermal barriers
developing closer to the root tip due to a reduction in the root
growth rate as well as accelerated maturation of the barriers
(Enstone et al., 2002). The observed reduced root growth rate
would presumably also lead to the xylem developing closer to
the root tip. However, the accelerated maturation rate of the
barriers could lead to the CS developing closer to the root tip
than the xylem (similar to the scenario shown in Figure 4E). This
could be a favorable adaptation to salt stress as it would result in
reduced passive uptake of ions (Figure 4G), although it would
also lead to reduced water uptake. Similarly, when comparing the
development of endodermal barriers across species or varieties
in order to explain differences in their transport abilities it would
seem important to consider the relative development of the xylem
and CS.
Altered development of the SL can be correlated with altered
development in the CS (Enstone et al., 2002). Hence, we
also explored scenarios representing this correlation. We found
similar behavior to that shown in Figure 4, with the additional
feature that any reduction in the total length of the SL resulted in
increased axial flow of ions in the xylem (results not shown).
In all of the scenarios considered, the peak uptake of
water and solutes into the root occurs in the region where
the xylem is present and either the CS is not present or is
still developing. This agrees with the experimentally observed
pattern of rapid uptake of water in the young region of the
root (Graham et al., 1974; Clarkson and Robards, 1975). In
contrast, the experimentally observed uptake of ions is more
variable (Lüttge, 1983). This variability is unsurprising due
to the variation in the types and locations of ion specific
transporters. However, for some ions, peak uptake has been
observed in this young region of the root; an example of
this is calcium (Robards et al., 1973; White, 2001). It is
worth noting that the modeled peak in ion uptake is not
significantly affected even if the ions have differing diffusive
Frontiers in Plant Science | www.frontiersin.org
FIGURE 4 | (A–F) Plots showing the lower section of the model root,
indicating the different initial positions of development of the CS and functional
xylem that were explored. Purple (yellow) vertical lines indicate the presence of
the CS (functional xylem). Dotted lines represent the gradual development of
the CS and xylem, while solid lines represent fully functioning CS and xylem.
Horizontal gray lines show the location of the start of the CS and xylem used in
the model simulations shown in all other figures. The arrows show the net
radial solute flux across the surface of the root at steady state for each
scenario, these flows are indicative of the solute fluxes into the xylem. The
same scale is used for the arrows across all six subfigures. Although these
arrows show solute fluxes, they are also indicative of the corresponding water
flow rates. The arrows shown represent the model results for Pb = −0.3 MPa.
(G) Line plots showing the steady state axial solute fluxes out of the top of the
model xylem as a function of the pressure boundary condition (representing
the transpiration rate) and the relative positions of the initial development of the
CS and xylem. The axial fluxes into the xylem compartment three cells down
from the top of the root were used to minimize the influence of the type of flux
boundary conditions chosen at the top of the root. The line colors and styles in
(G) match the colors and styles of the boxes around the corresponding
scenario in (A–F). All simulations assume 100 mM of 2 monovalent cations
and 2 monovalent anions in the soil. All ions are assumed to have identical
transport properties and the transport parameters used in the simulations are
provided in Table 1.
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outside the endodermal barriers. Overall these results point to the
importance of considering the effect of the level of transpiration
when experimentally investigating the function of endodermal
barriers, particularly when using the presence of a build up of
ions outside the CS as an experimental measure of how well it is
functioning as a barrier (e.g., Lauchli et al., 2008).
In contrast to the results in the DZ, the steady state
concentrations in the EZ are identical and equal to the ion
concentrations in the soil for both transpiration scenarios (see
Figures 5A–C). Since there is only negligible convection in the
EZ, transpiration conditions do not influence the uptake of
ions in this hydraulically isolated region of the root. Similarly,
for both transpiration scenarios the concentration of ions in
the xylem and pericycle decreases through the transition zone
from the EZ to the DZ due to the development of a functional
xylem . However, this decrease is more pronounced under higher
transpiration conditions (see Figures 5A–C).
The effect of passage cells on the distribution of ions extends
across several tissue regions (see Figure 5C). There are higher
levels of both diffusion and convection across the endodermis in
the vicinity of passage cells compared to the surrounding cells
which are affected by the SL (Figures 5D,E). Where there are
passage cells, ions build up in the pericycle as both diffusion and
convection draw ions in from the surrounding outer tissues (see
the yellow diffusive and red convective arrows in Figures 5D,E).
Again, these results confirm the function of passage cells as key
entry points into the stele (Peterson and Enstone, 1996).
The arrows in Figures 5A,B show that the pattern of ion
uptake is not qualitatively affected by the level of transpiration.
However, the magnitude of ion uptake is strongly dependent on
the rate of transpiration.
The significant differences in the distribution of ions for the
different transpiration conditions are supported by experimental
observations. For example, Møller et al. (2009) found that
the measured radial distribution of Na+ and K+ differed
substantially between plants grown in conditions where there was
little or no transpiration and those plants grown in transpiring
conditions. In addition, our predicted increase in the magnitude
of water uptake along the length of the root for higher
transpiration conditions (Figures 5A,B) has also been observed
experimentally (e.g., Sanderson, 1983).
permeabilities (results not shown). Our results show that the
location of the peak in uptake is less obvious if the CS extends
beyond the xylem (compare Figures 4E,F to Figures 4C,D) or
if the transpiration level is low (see the next section). Hence,
differences in the anatomy of plant roots as well as experimental
conditions may affect the extent and prominence of the peak in
uptake.
The model results in Figures 4A–F show the locations of peak
uptake. However, they do not specify which region of the root
contributes the most to the total uptake of ions and water. Due
to the longer length of the mature region of the root compared
to the short length of the high uptake region, the mature region
may contribute more to the overall uptake of ions and water. The
presence of lateral roots also contributes to the uptake in the more
mature regions of the root.
Low vs. High Transpiration Leads to
Significantly Different Spatial Distributions
of Ions in the DZ, but Not in the EZ
Figures 3, 4G initiated a study of the effect of transpiration on the
rate of ion and water uptake. In this section we explore in greater
detail the influence of transpiration on the distribution of ions
within the root (Figure 5). The effect of varying the transpiration
level is particularly felt in the DZ, with the level of transpiration
influencing both the axial and radial distribution of ions.
The qualitative behavior of the axial distribution of ions in the
DZ varies depending on the level of transpiration. Under high
transpiration there is a steady decline in concentration in the
xylem along the length of the root in the DZ (Figure 5B and
red dotted line in Figure 5C), whereas under low transpiration
conditions the ion concentrations in the xylem increase along
the length of the root in the DZ (Figure 5A and solid red line
in Figure 5C). This dichotomy arises from the competing effects
of diffusion and convection. High transpiration results in ions
being convected away faster than can be replaced by diffusion,
while under low transpiration the low level of convection cannot
offset the influx of ions by diffusion into the xylem from the
surrounding tissues.
In terms of the radial distribution of ions in the DZ, the
concentration of ions in the xylem and pericycle is lower at
higher levels of transpiration (Figure 5B and dotted lines in
Figure 5C compared to Figure 5A and solid lines in Figure 5C)
due to greater convection up and out of the root. In addition, at
higher levels of transpiration there is a build up of ions in the
endodermis at locations where only the CS is present (including
at passage cell locations), while this build up is not evident at
lower transpiration levels (green dotted lines compared to solid
lines in Figure 5C). Figures 5D,E highlight how the interactions
between diffusion and convection generate these contrasting
results. In particular, where only the CS is present, a high level
of convection (red arrows in Figure 5E) is required to offset
the process of diffusion, which counteracts the formation of
any concentration gradients (see yellow arrows in Figure 5E). In
contrast, at low levels of transpiration, the inward convection of
ions is small or even reversed due to osmotic pressure gradients
(red arrows in Figure 5D) and hence there is no build up of ions
Frontiers in Plant Science | www.frontiersin.org
Measurements of Transient vs. Steady
State Ion Fluxes Reflect Different Aspects
of Root Function
The design of experiments to measure the fluxes of ions and/or
water across root surfaces (e.g., Zhou et al., 2011) should
incorporate an appropriate time frame to match the purpose
of the experiment, i.e., are transient or steady state conditions
appropriate? Transient ion fluxes demonstrate which regions
along the length of the root fill up most rapidly, while steady
state fluxes indicate the regions through which the main uptake
occurs over time. This is highlighted in Figure 6, which shows
how the ion concentrations (colormap) and the radial solute
fluxes (arrows) change over time. For clarity only the net radial
solute fluxes into the epidermis and pericycle are shown in
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FIGURE 5 | (A,B) Plots of steady state ion concentrations (colormaps) and ion fluxes (arrows) for two different transpiration conditions: Pb = 0 MPa and Pb = −0.5
MPa respectively. The arrows are also indicative of water flow rates. The same scale is used for the arrows in (A,B). (C) Line plots of ion concentrations in (A) (solid
lines) and (B) (dotted lines). The line colors refer to tissue regions: xylem (red lines), pericycle (blue lines) and endodermis (green lines). (D,E) show zoomed-in sections
of results in (A,B), respectively, at the location of the passage cell indicated by asterisks. Yellow arrows on the left show the diffusive component of the total ion flow,
red arrows on the right show the convective component (adding the two together gives the total flow). The arrows are shown in the center of their corresponding
model compartment. The arrows in (D,E) are drawn to different scales for clarity, an equal length arrow in (E) represents a 10 times larger flux than in (D). The black
arrows highlight the convective flow path and are not drawn to scale. All simulations assume 100 mM of 2 monovalent cations and 2 monovalent anions in the soil. All
ions are assumed to have identical transport properties and the transport parameters used in the simulations are provided in Table 1.
fluxes across the root surface are representative of where the
main uptake of ions into the inner regions occurs. This is
provided there are no large vertical fluxes in the intervening
regions. Further complications in flux measurements would
arise if the ions being measured are converted to other forms
before reaching the stele (e.g., the reduction of nitrate to
nitrite).
Figure 6, rather than the fluxes into all of the root tissues. As
in previous figures, the arrows in Figure 6 also represent the
water flow rates, which are in direct proportion to the ion
fluxes. The results indicate that the solute fluxes at the surface
are not representative of those in the inner regions of the
root until the system is at or near steady state (Figures 6A–C).
However, when the system is at steady state (Figure 6D), solute
Frontiers in Plant Science | www.frontiersin.org
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FIGURE 6 | Plots of transient ion concentrations (colormap) and ion fluxes (arrows) for the root structure shown in Figure 2D. For clarity only the net
radial solute fluxes across the root surface (into the epidermis) and across the endodermis-pericycle interface are shown. The initial condition for this simulation was
the steady state result for a simulation carried out with 100 mM of 1 monovalent cation and 1 monovalent anion in the soil (e.g., KNO3 ). The simulation was then
carried out assuming 100 mM of 2 monovalent cations and 2 monovalent anions in the soil (e.g., the existing KNO3 with NaCl added). The results shown are for the
added ions. The results show the changes over time after the ions were added: (A) 15 min after, (B) 2 h after, (C) 4 h after, (D) at steady state (approximately 1 day
after). These simulations were carried out using Pb = −0.5 MPa. All ions are assumed to have identical transport properties and the transport parameters used in the
simulations are provided in Table 1.
4. DISCUSSION
EZ at steady state is likely to be driven by active processes as
there is only minimal passive uptake in this region (see arrows
in Figures 2D, 6D). Similarly, any significant uptake of water in
this region is likely to be driven by osmotic pressure gradients
developed by the active transport of ions.
It is no surprise that the level of transpiration has a very
significant effect on the rate of ion and water uptake (Figures 3,
4G and arrows in Figure 5), as well as on the spatial distribution
of ions radially and axially inside the DZ of the root (Figure 5).
Indeed, the level of impact that transpiration has on the
spatial distribution of ions is of a similar scale to the level of
impact of the presence of the endodermal barriers (compare
Figures 2, 5). Our results support the experimental findings
of Møller et al. (2009) that different transpiration conditions
lead to differences in the radial distribution of ions across
plant roots. In addition, our results highlight the fact that
some transport behavior becomes more distinct as the level of
transpiration increases, e.g., the influence of the endodermal
Our work focuses on the passive uptake of ions and water, in
particular, exploring the interaction between the processes of
diffusion and convection. The results therefore show the ion
uptake patterns and concentration distributions that can be
explained by passive processes and their interaction with cell
differentiation. However, using these results one can also infer
information about the active transport of ions. In particular, our
results describe the pattern of ion and water uptake that occurs
when only passive processes operate (arrows in Figure 2D) and
highlight that qualitatively this pattern is relatively unaffected
by either differences in the level of transpiration (arrows in
Figures 5A,B) or differences in cell development (Figures 4A–F),
although the magnitudes can be substantially affected. Hence, an
ion uptake pattern that qualitatively differs substantially from
Figure 2D would indicate the operation of an active transport
mechanism. For example, any significant uptake of ions in the
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to the modeled influx found at the location of passage cells (e.g.,
Figure 2D).
In this paper we have explored passive transport processes.
However, to investigate the uptake and transport of specific
ions (such as K+ or Na+ ) it is necessary to incorporate models
of active transport, such as the model we have developed
recently (Foster and Miklavcic, 2015). Such a challenge requires
the explicit modeling of symplastic and apoplastic pathways,
which would allow us to investigate more specific aspects
of cell differentiation that cannot be included in the current
model. For example, one of the suspected functions of the
CS is to prevent backflow of ions out of the stele under low
transpiration conditions (Enstone et al., 2002). An investigation
of this process would require active transport mechanisms to be
included since it is thought that active transport is responsible
for the build up of ions in the xylem under low or nontranspiring conditions (Enstone et al., 2002). In addition, a more
complex model would allow a clearer separation of the function
of the CS (which blocks the apoplastic pathways) from the
function of the SL (which is thought to block uptake into the
symplastic pathway), including a further exploration of the role
of passage cells.
barriers on ion uptake (Figure 3), as well as the positions of
peak uptake (Figures 5A,B). Hence, our findings suggest that
the results of ion and water measurements conducted on nontranspiring plants or excised roots (with no suction applied to
cut end) are likely to differ substantially from measurements
conducted on intact, transpiring plants. In particular, the effects
of convective processes will be more obvious under higher
transpiring conditions, whereas under low or non-transpiring
conditions the effects of active processes are likely to be more
obvious, although under all growth conditions the transport
will also be affected by diffusion. These differences need to
be considered when attempting to compare the results of
experiments conducted under different transpiration conditions
(see e.g., Møller and Tester, 2007).
The effects of the transpiration level also have ramifications
for other modeling efforts. Firstly, our more detailed simulations
of water transport concur with previous modeling findings that
the level of convection is a significant factor in determining
the radial distribution of ions (Claus et al., 2013). Convection
is clearly an essential mechanism to include in models of
ion transport in roots in which functional xylem is present.
Secondly, the distinctly different patterns of ion distribution
resulting from differing transpiration levels (Figures 5A–C)
suggest the importance of investigating the interactions between
convective and membrane transport processes in any study of
the effects of membrane transporters on the pattern of solute
distribution (e.g., Claus et al., 2013; Sakurai et al., 2015). Thirdly,
convection does not seem to be significant in regions of root
where the xylem is not conductive, such as in the EZ. In
our case, the zone of influence of the xylem extended only
a few cells in the direction of the root tip from the initial
conductive cells. These findings support earlier assumptions that
it is unnecessary to include convection when modeling ion
transport in regions where the xylem is not functional (e.g.,
Shimotohno et al., 2015). Hence, while it is not necessary for
convection to be incorporated into models of ion transport in
the meristematic zone and the EZ, it may be very important for
convection to be considered in the more mature regions of the
root.
It has been suggested that passage cells function as low
resistance pathways for the uptake of water and at least some
ions (Peterson and Enstone, 1996). Our findings (Figures 2D,
3, 5) give this position some support. This role has been
suggested due to the consistent positioning of passage cells
adjacent to protoxylem poles (Peterson and Enstone, 1996).
However, there is currently very limited direct experimental
evidence identifying the function of passage cells (Andersen et al.,
2015). To investigate the role of passage cells further it would be
necessary to differentiate between the apoplastic and symplastic
pathways.
Our efforts to model transport within a single root do not
include contributions from lateral roots to the uptake of ions
and water. Not only do lateral roots increase the root surface
area available for uptake, their formation also interrupts the CS
(Vermeer et al., 2014), potentially allowing leakage of ions and
water across the endodermis. Hence, the influx of ions and water
at the location of lateral root formation would likely be similar
Frontiers in Plant Science | www.frontiersin.org
5. CONCLUSIONS
In this paper we have presented the results of simulations of the
transport of ions and water in a plant root via passive transport
processes. The model root incorporates both different tissue types
and different developmental zones. As in our previous works
(Foster and Miklavcic, 2013, 2014), the model includes the selfconsistent interaction between the transport of ions and water.
We have used the model to simulate a wide range of transport
scenarios and found that, in all instances where the endodermal
barriers were present, the peak uptake of water and ions occurred
at the start of the DZ. In addition, we found that there was
no substantial uptake in the EZ at steady state due to the lack
of functional xylem in this region. From this we infer that any
observed uptake of ions (or water) in this developmental zone
under steady state conditions is likely to be due to active transport
processes. The results also highlight that the level of transpiration
has a significant impact on water and ion transport in the DZ.
This should be taken into consideration both when conducting
experiments and developing models to examine transport in this
developmental zone.
We have used our model of passive transport to infer
information about active transport processes. However, more
detailed modeling is required to examine the transport of specific
ions. For example, combining a single cell model of active
and passive transport of ions as well as water, which we have
developed previously (Foster and Miklavcic, 2015), with the root
model presented here would allow more detailed exploration
of ion transport (specifically Na+ , K+ , and Cl− ) across the
different developmental zones. For example, it would allow an
examination of how each of the endodermal barriers influence
the different transport pathways. This is the subject of ongoing
efforts. Nevertheless, the passive processes of diffusion and
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convection modeled in this paper are fundamental to ion uptake.
Hence, the results discussed in this paper establish a baseline of
transport phenomenon on which active transport mechanisms
are imposed and with which active transport mechanisms
interact.
FUNDING
AUTHOR CONTRIBUTIONS
SUPPLEMENTARY MATERIAL
The authors were equal contributors to the design of simulations,
analysis of results and drafting of the paper. KF performed the
simulations.
The Supplementary Material for this article can be found
online at: http://journal.frontiersin.org/article/10.3389/fpls.2016.
00914
This project is supported by an Australian Postgraduate Award
and a Grains Industry Research Scholarship from the Grains
Research and Development Corporation for KF.
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Conflict of Interest Statement: The authors declare that the research was
conducted in the absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
The reviewer ID and handling Editor declared their shared affiliation, and
the handling Editor states that the process nevertheless met the standards of a fair
and objective review.
Copyright © 2016 Foster and Miklavcic. This is an open-access article distributed
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